which is to be interpreted in the convolution algebra \(\mathcal {B}(L^2({\mathbb {R}}^3), \mathcal {M}_t L^2({\mathbb {R}}^3))\)  where \(\mathcal {M}_t\)  are the complex measures on the line.
On the other hand,
\(\widehat{T_{V_2, V_1} }(\lambda ) = iV_2 R_0^{-}(\lambda ) V_1\)  with \( R_0^{-}(\lambda ) = (H_0 - (\lambda -i0))^{-1}\) .
Hence, the invertibility of \(\delta _0\mathrm {Id}-i T_{V_2, V_1}\)  in \(\mathcal {B}(L^2({\mathbb {R}}^3), \mathcal {M}_t L^2({\mathbb {R}}^3))\)  is the same as the pointwise invertibility of the Birman-Schwinger operator
\(\mathrm {Id}+V_2 R_0^{-}(\lambda ) V_1\) . This equivalence is delicate, and requires \(V\in L^{\frac{3}{2},1}({\mathbb {R}}^3)\)  the Lorentz space, whence \(V_1, V_2\in L^{3,2}({\mathbb {R}}^3)\) , and also the Keel-Tao Strichartz endpoint [1]}. For the abstract Wiener theorem in this context, see [2]} and [3]}.
